The Hodograph Equation for Slow and Fast Anisotropic Interface Propagation

Using the model of fast phase transitions and previously reported equation of the Gibbs-Thomson-type, we develop an equation for the anisotropic interface motion of the Herring-Gibbs-Thomson-type. The derived equation takes the form of a hodograph equation and in its particular case describes motion by mean interface curvature, the relationship 'velocity - Gibbs free energy', Klein-Gordon and Born-Infeld equations related to the anisotropic propagation of various interfaces. Comparison of the present model predictions with the molecular-dynamics simulation data on nickel crystal growth (obtained by Jeffrey J. Hoyt et al. and published in Acta Mater. 47 (1999) 3181) confirms the validity of the derived hodograph equation as applicable to the slow and fast modes of interface propagation. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'. © 2021 The Authors. Data accessibility. Electronic supplementary material on asymptotic analysis of the hyperbolic phase field model are attached to the main text of the manuscript. Authors’ contributions. Both authors contributed to the derivation, analytical treatments and analysis of results. A.S. carried out calculations for comparison of the analytical equation with data of molecular dynamics simulations. Competing interests. We declare we have no competing interests. Funding. The funding has been made for P.K.G. by German Science Foundation (DFG-Deutsche Forschungsgemeinschaft) under the Project GA 1142/11-1. Acknowledgements. Authors thank Jeffrey J. Hoyt for his valuable explanations about molecular dynamic simulation data of Ni. P.K.G. acknowledges financial support of German Science Foundation (DFG-Deutsche Forschungsgemeinschaft). A.S. thanks M. Bennai for hosting the present work in the research activities of LPMC.